Optimal. Leaf size=193 \[ \frac{2 (35 A+31 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 a d}-\frac{4 (35 A+37 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A+C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right )}{\sqrt{a} d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}-\frac{2 C \sin (c+d x) \cos ^2(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.559901, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3046, 2983, 2968, 3023, 2751, 2649, 206} \[ \frac{2 (35 A+31 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 a d}-\frac{4 (35 A+37 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A+C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right )}{\sqrt{a} d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}-\frac{2 C \sin (c+d x) \cos ^2(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 3046
Rule 2983
Rule 2968
Rule 3023
Rule 2751
Rule 2649
Rule 206
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx &=\frac{2 C \cos ^3(c+d x) \sin (c+d x)}{7 d \sqrt{a+a \cos (c+d x)}}+\frac{2 \int \frac{\cos ^2(c+d x) \left (\frac{1}{2} a (7 A+6 C)-\frac{1}{2} a C \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{7 a}\\ &=-\frac{2 C \cos ^2(c+d x) \sin (c+d x)}{35 d \sqrt{a+a \cos (c+d x)}}+\frac{2 C \cos ^3(c+d x) \sin (c+d x)}{7 d \sqrt{a+a \cos (c+d x)}}+\frac{4 \int \frac{\cos (c+d x) \left (-a^2 C+\frac{1}{4} a^2 (35 A+31 C) \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{35 a^2}\\ &=-\frac{2 C \cos ^2(c+d x) \sin (c+d x)}{35 d \sqrt{a+a \cos (c+d x)}}+\frac{2 C \cos ^3(c+d x) \sin (c+d x)}{7 d \sqrt{a+a \cos (c+d x)}}+\frac{4 \int \frac{-a^2 C \cos (c+d x)+\frac{1}{4} a^2 (35 A+31 C) \cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx}{35 a^2}\\ &=-\frac{2 C \cos ^2(c+d x) \sin (c+d x)}{35 d \sqrt{a+a \cos (c+d x)}}+\frac{2 C \cos ^3(c+d x) \sin (c+d x)}{7 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (35 A+31 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{105 a d}+\frac{8 \int \frac{\frac{1}{8} a^3 (35 A+31 C)-\frac{1}{4} a^3 (35 A+37 C) \cos (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx}{105 a^3}\\ &=-\frac{4 (35 A+37 C) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}-\frac{2 C \cos ^2(c+d x) \sin (c+d x)}{35 d \sqrt{a+a \cos (c+d x)}}+\frac{2 C \cos ^3(c+d x) \sin (c+d x)}{7 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (35 A+31 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{105 a d}+(A+C) \int \frac{1}{\sqrt{a+a \cos (c+d x)}} \, dx\\ &=-\frac{4 (35 A+37 C) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}-\frac{2 C \cos ^2(c+d x) \sin (c+d x)}{35 d \sqrt{a+a \cos (c+d x)}}+\frac{2 C \cos ^3(c+d x) \sin (c+d x)}{7 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (35 A+31 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{105 a d}-\frac{(2 (A+C)) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,-\frac{a \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{d}\\ &=\frac{\sqrt{2} (A+C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a+a \cos (c+d x)}}\right )}{\sqrt{a} d}-\frac{4 (35 A+37 C) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}-\frac{2 C \cos ^2(c+d x) \sin (c+d x)}{35 d \sqrt{a+a \cos (c+d x)}}+\frac{2 C \cos ^3(c+d x) \sin (c+d x)}{7 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (35 A+31 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{105 a d}\\ \end{align*}
Mathematica [A] time = 0.320768, size = 89, normalized size = 0.46 \[ \frac{2 \cos \left (\frac{1}{2} (c+d x)\right ) \left (105 (A+C) \tanh ^{-1}\left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-2 \sin ^3\left (\frac{1}{2} (c+d x)\right ) (70 A+24 C \cos (c+d x)+15 C \cos (2 (c+d x))+101 C)\right )}{105 d \sqrt{a (\cos (c+d x)+1)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 253, normalized size = 1.3 \begin{align*}{\frac{1}{105\,d}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sqrt{a \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( -240\,C\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}+336\,C\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}-140\,\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \left ( A+2\,C \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+105\,\sqrt{2}\ln \left ( 4\,{\frac{\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+a}{\cos \left ( 1/2\,dx+c/2 \right ) }} \right ) aA+105\,\sqrt{2}\ln \left ( 4\,{\frac{\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+a}{\cos \left ( 1/2\,dx+c/2 \right ) }} \right ) aC \right ){a}^{-{\frac{3}{2}}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{a \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70162, size = 485, normalized size = 2.51 \begin{align*} \frac{4 \,{\left (15 \, C \cos \left (d x + c\right )^{3} - 3 \, C \cos \left (d x + c\right )^{2} +{\left (35 \, A + 31 \, C\right )} \cos \left (d x + c\right ) - 35 \, A - 43 \, C\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right ) + \frac{105 \, \sqrt{2}{\left ({\left (A + C\right )} a \cos \left (d x + c\right ) +{\left (A + C\right )} a\right )} \log \left (-\frac{\cos \left (d x + c\right )^{2} - \frac{2 \, \sqrt{2} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt{a}} - 2 \, \cos \left (d x + c\right ) - 3}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right )}{\sqrt{a}}}{210 \,{\left (a d \cos \left (d x + c\right ) + a d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.76573, size = 213, normalized size = 1.1 \begin{align*} -\frac{\frac{105 \, \sqrt{2}{\left (A + C\right )} \log \left ({\left | -\sqrt{a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} \right |}\right )}{\sqrt{a}} + \frac{4 \,{\left ({\left (\sqrt{2}{\left (35 \, A a^{3} + 46 \, C a^{3}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 14 \, \sqrt{2}{\left (5 \, A a^{3} + 4 \, C a^{3}\right )}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 35 \, \sqrt{2}{\left (A a^{3} + 2 \, C a^{3}\right )}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3}}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a\right )}^{\frac{7}{2}}}}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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